Because quadruples of collinear points are not freely transitive under the projective group, a fixed collinear quadruple can serve as a unit of measurement

Klein’s insight, following von Staudt, was that an exactly similar argument involving quadruples of collinear points can be used to define cross-ratio in projective geometry. The projective group preserves straight lines, and any ordered triple of collinear points can be mapped to any ordered triple of collinear points, and the map that sends a given ordered triple of distinct points to another ordered triple of distinct points is unique, but there is no transformation in the group that can map